Problem 107 - College Algebra

Question

Which one of the following is true? a. \(4^{-2}<4^{-3}\) b. \(5^{-2}>2^{-5}\) c. \((-2)^{4}=2^{-4}\) d. \(5^{2} \cdot 5^{-2}>2^{5} \cdot 2^{-5}\)

Step-by-Step Solution

Answer

The true statements among the given options are b and d.

Step 1: Comparing option a

Evaluate each expression: \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\) and \(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\). \(\frac{1}{16} > \frac{1}{64}\), therefore option a is false.

Step 2: Comparing option b

Evaluate each expression: \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\) and \(2^{-5} = \frac{1}{2^5} = \frac{1}{32}\). \(\frac{1}{25} > \frac{1}{32}\), therefore option b is true.

Step 3: Comparing option c

Evaluate each expression: \((-2)^4 = 16\) and \(2^{-4} = \frac{1}{2^4} = \frac{1}{16}\). 16 ≠ \(\frac{1}{16}\), therefore option c is false.

Step 4: Comparing option d

Evaluate each expression: \(5^2 \cdot 5^{-2} = 5^{(2-2)} = 5^0 = 1\) and \(2^{5} \cdot 2^{-5} = 2^{(5-5)} = 2^0 = 1\). 1 = 1, therefore option d is true.

Key Concepts

Negative Exponents

Understanding negative exponents is crucial to mastering algebra. Any number raised to a negative exponent is equivalent to 1 divided by that same number raised to the positive of the exponent. Simply put, for any nonzero number 'a' and positive integer 'n', the expression \(a^{-n}\) equals \(\frac{1}{a^n}\).

For instance, the expression \(4^{-2}\) is the same as \(\frac{1}{4^2}\), which simplifies to \(\frac{1}{16}\). This concept allows us to transform negative exponents into positive ones, facilitating easier comparison and manipulation of terms.

Exponent Properties

Exponent properties, or laws of exponents, are sets of rules that describe how to handle expressions involving exponents. A few fundamental properties include the product of powers property \(a^m \cdot a^n = a^{m+n}\), the power of a power property \((a^m)^n = a^{m \cdot n}\), and the power of a product property \((ab)^n = a^n \cdot b^n\).

These rules are essential tools that allow us to simplify and solve a wide variety of algebraic expressions. In our exercise example, using the product of powers property, we simplify \(5^2 \cdot 5^{-2}\) to \(5^{2-2} = 5^0 = 1\), illustrating the simplification of complex expressions.

Comparing Exponents

When comparing exponents, it is important to consider both the base number and the exponent value. For the same base number, a smaller negative exponent actually represents a larger quantity, as you are taking the reciprocal of a smaller number. This can sometimes seem counterintuitive at first glance.

Consider our textbook exercise where we compare \(4^{-2}\) and \(4^{-3}\). Even though -3 is less than -2, \(4^{-3}\), which is \(\frac{1}{64}\), is actually smaller than \(4^{-2}\), which is \(\frac{1}{16}\). Understanding how to appropriately compare exponents allows for correct evaluation and ordering of exponential expressions.

Exponential Expressions

Exponential expressions are a way to represent repeated multiplication of the same number. They consist of a base and an exponent, demonstrating how many times the base is multiplied by itself. Exponential notions are not just limited to positive integers. They include zero, negative numbers, fractions, and even irrational numbers as exponents.

When an exponent is zero, as seen with \(5^0\) or \(2^0\), the value of the expression is always 1, regardless of the base (as long as the base is not zero). In algebra, working with exponential expressions involves applying laws of exponents and accurately operating within their defined properties to simplify and solve.